Does the father bear the sin for his wife holding their daughter’s birthday party at her family’s home, given his prior knowledge and objection to it?

You have done well in clarifying the ruling on celebrating birthdays for your wife and your refusal to hold such a celebration, and that her holding this celebration in her parents' home constitutes disobedience to your command. There is no blame on you for what your wife did as long as you were unable to prevent her, because changing an evil is contingent upon ability. There is no harm in speaking to your wife's family and explaining to them that it is impermissible for them to aid their daughter in disobeying her husband. It is also advisable that you explain to your wife the importance of obeying her husband.